7/25/2019 Amo 2014 Paper

http://slidepdf.com/reader/full/amo-2014-paper 1/2

THE 2014 AUSTRALIAN MATHEMATICAL OLYMPIAD

A USTRAL IAN  MATHEMATICAL  O LYMPIAD  COMMITTEE

A   DEPARTMENT  OF  THE  AUSTRAL IAN   MATHEMAT ICS  TRUST

A USTRAL IAN  MATHEMAT ICS T RUSTThe Mathematics Olympiads are supported by the

Australian Government Department of Education through

the Mathematics and Science Participation Program.

DAY 1

Tuesday, 11 February 2014

Time allowed: 4 hours

No calculators are to be used.

Each question is worth seven points.

1.   The sequence  a1, a2, a3, . . .  is defined by  a1 = 0 and, for  n ≥ 2,

an = maxi=1,...,n−1

i + ai + an−i

.

(For example,  a2 = 1 and  a3 = 3.)

Determine  a200.

2.   Let AB C  be a triangle with  ∠BAC < 90◦. Let k  be the circle through  A that is tangent

to BC  at  C . Let  M  be the midpoint of  BC , and let  AM   intersect  k  a second time at  D.

Finally, let  BD  (extended) intersect  k  a second time at  E .

Prove that  ∠BAC  = ∠CAE .

3.   Consider labelling the twenty vertices of a regular dodecahedron with twenty different

integers. Each edge of the dodecahedron can then be labelled with the number  |a − b|,

where  a  and  b  are the labels of its endpoints. Let  e  be the largest edge label.

What is the smallest possible value of  e  over all such vertex labellings?

(A regular dodecahedron is a polyhedron with twelve identical regular pentagonal faces.)

4.   Let  N+ denote the set of positive integers, and let  R  denote the set of real numbers.

Find all functions  f   : N+ → R that satisfy the following three conditions:

(i)   f (1) = 1,

(ii)   f (n) = 0 if  n  contains the digit 2 in its decimal representation,

(iii)   f (mn) = f (m)f (n) for all positive integers  m, n.

c  2014 Australian Mathematics Trust

7/25/2019 Amo 2014 Paper

http://slidepdf.com/reader/full/amo-2014-paper 2/2

THE 2014 AUSTRALIAN MATHEMATICAL OLYMPIAD

A USTRAL IAN  MATHEMATICAL  O LYMPIAD  COMMITTEE

A   DEPARTMENT  OF  THE  AUSTRAL IAN   MATHEMAT ICS  TRUST

A USTRAL IAN  MATHEMAT ICS T RUSTThe Mathematics Olympiads are supported by the

Australian Government Department of Education through

the Mathematics and Science Participation Program.

DAY 2

Wednesday, 12 February 2014

Time allowed: 4 hours

No calculators are to be used.

Each question is worth seven points.

5.  Determine all non-integer real numbers  x  such that

x + 2014

x  = x +

 2014

x  .

(Note that x denotes the largest integer that is less than or equal to the real number  x.

For example,  20.14 = 20 and  −20.14 =  −21.)

6.   Let  S  be the set of all numbers

a0 + 10 a1 + 102 a2 + · · · + 10n an   (n = 0, 1, 2, . . .)

where

(i)   ai  is an integer satisfying 0 ≤  ai ≤  9 for  i = 0, 1, . . . , n and  an = 0, and

(ii)   ai  < ai−1 + ai+1

2  for  i = 1, 2, . . . , n− 1.

Determine the largest number in the set  S .

7.   Let  ABC  be a triangle. Let  P   and  Q  be points on the sides  AB   and  AC , respectively,

such that  BC  and  PQ are parallel. Let  D  be a point inside triangle  APQ. Let  E  and  F 

be the intersections of  PQ  with  BD   and  CD, respectively. Finally, let OE   and  OF   be

the circumcentres of triangle  DEQ and triangle  DFP , respectively.

Prove that  OE OF   is perpendicular to  AD.

8.   An  n× n  square is tiled with 1 × 1 tiles, some of which are coloured. Sally is allowed to

colour in any uncoloured tile that shares edges with at least three coloured tiles. She

discovers that by repeating this process all tiles will eventually be coloured.

Show that initially there must have been more than   n2

3  coloured tiles.

c  2014 Australian Mathematics Trust